Chemical potential relation with osmotic pressure

Osmotic pressure measurements can be used to evaluate x at small volume fractiOTis of polymer. The osmotic pressure n of a solution relative to pure solvent is related to the chemical potential and, with Eq. (14.8), to y through the thermodynamic expression... 

Although the kinetics of liquid uptake to attain gel-saturation is history-dependent, the composition at the true end-state (i.e. thermodynamic equilibrium in excess liquid) is not therefore the observed end-state is usually reproducible [19]. Gel-saturation is attained when the restraining force (per unit area) of the polymeric crosslinked network becomes equal and opposite to the osmotic pressure that causes the system to swell [20], In other words saturation is achieved when the chemical potential of swelling liquid, p1 in the swollen network is equal to the chemical potential of the excess pure liquid, p , outside the network. It was logical to anticipate that the volume of liquid sorbed per gram of polymer, at this state of thermodynamic equilibrium with excess liquid, would correlate with the molecular structure of the liquid. In fact two parameters already exist which relate the sorption affinity to the molecular structure, namely the solubility parameter, 8, first proposed by Hildebrand [21], and the interaction parameter, %, introduced by Flory [22] and Huggins [23-26],... 

The starting point of McMillan-Mayer theory is a relationship between distribution functions at different activity sets. The derivation of this relationship is the difficult part of the theory. But once obtained, the relation leads to an expression for the osmotic pressure of a solution, since the components permeable to the osmotic membrane have the same chemical potential on both sides of the membrane while those impermeable have differing chemical potentials. A lengthy computation then leads to an expansion for the osmotic pressure, completely analogous to the activity expansion of the pressure in the theory of imperfect gases. Indeed, for the purpose of comparing gas theory with solution theory, it helps to regard the gas as a solute in a very special and very simple solvent— vacuum. The X expansion is. 

Measurements performed to determine the molar masses of polymers yield - as a valuable byproduct - information on the pair interaction between the macromolecules [30]. The composition dependence of the osmotic pressure Tiosm observed via membrane osmometry is directly related to the chemical potential of the solvent [cf. (14) of Sect. 2] and provides the second osmotic virial coefficient A2, from which Xo> Ihe Flory-Huggins interaction parameter in the limit of high dilution becomes accessible [cf. (15)]. Such data are particularly valuable because they can be measured with higher accuracy than the x values for concentrated polymer solutions and because they represent a solid starting point for the sometimes very complex function xiV )- In principle, membrane osmometry can also be operated with polymer solutions of different composition in the two chambers (differential osmometry) to gain data for higher polymer concentrations however, little use is made of this option. 

If a polymer solution is modeled by an assembly of self-avoiding walks on a lattice, a basic physical quantity is the osmotic pressure II. Carrying out a simulation with a fixed number JT of chains of length at a lattice of volume V with one of the dynamic algorithms described in Section 1.2.2, the osmotic pressure is not straightforward to sample. If one had methods that yielded the excess chemical potential and the Helmholtz free energy Fjr, one would find II from the thermodynamic relation... 

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